Questions: (x+3)^2 + (y+4)^2 = 1

Transcript text: $(x+3)^{2}+(y+4)^{2}=1$

Solution

Solution Steps

To express $ y $ as a function of $ x $ from the given equation of a circle, we need to solve for $ y $ in terms of $ x $. This involves isolating $ y $ on one side of the equation.

Solution Approach

Start with the given equation of the circle: $(x+3)^2 + (y+4)^2 = 1$.
Isolate $(y+4)^2$ by subtracting $(x+3)^2$ from both sides.
Take the square root of both sides to solve for $ y + 4 $.
Finally, isolate $ y $ by subtracting 4 from both sides.

Step 1: Start with the Given Equation

We start with the given equation of the circle: \[ (x+3)^2 + (y+4)^2 = 1 \]

Step 2: Isolate $(y+4)^2$

Subtract $(x+3)^2$ from both sides: \[ (y+4)^2 = 1 - (x+3)^2 \]

Step 3: Simplify the Right-Hand Side

Simplify the expression on the right-hand side: \[ (y+4)^2 = 1 - (x+3)^2 = 1 - (x^2 + 6x + 9) = -x^2 - 6x - 8 \]

Step 4: Take the Square Root of Both Sides

Take the square root of both sides to solve for $y + 4$: \[ y + 4 = \pm \sqrt{-(x^2 + 6x + 8)} \]

Step 5: Simplify the Square Root Expression

Simplify the expression under the square root: \[ y + 4 = \pm \sqrt{-(x+2)(x+4)} \]

Step 6: Isolate $y$

Subtract 4 from both sides to isolate $y$: \[ y = -4 \pm \sqrt{-(x+2)(x+4)} \]

Final Answer

The solutions for $y$ as a function of $x$ are: \[ \boxed{y = -4 + \sqrt{-(x+2)(x+4)}} \quad \text{and} \quad \boxed{y = -4 - \sqrt{-(x+2)(x+4)}} \]

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